Advanced Calculus Single Variable

6.10 Sequences Of Polynomials, Weierstrass Approximation

It turns out that if f is a continuous real valued function defined on an interval,

[a,b]

then
there exists a sequence of polynomials,

{pn}

such that the sequence converges uniformly to f
on

[a,b]

. I will first show this is true for the interval

[0,1]

and then verify it is true on any
closed and bounded interval. First here is a little lemma which is interesting for its
own sake in probability. It is actually an estimate for the variance of a binomial
distribution.